Optimal Initial Perturbations Research Articles

Optimal initial duty for improved MPPT under change in irradiances and partial shading conditions

Abstract Solar photovoltaic cells are widely used in renewable energy power generation. The efficient operation of these photovoltaic (PV) system is based on maximum power point tracking (MPPT). Various MPPT methods have been proposed to increase the output of a PV system. But confusion arises when choosing the optimal MPPT algorithm and its parameters such as perturbation time and initial duty. This paper examines the optimal initial duty and perturbation time to improve the performance of conventional MPPT techniques under uniform shading and partial shading. During the change in irradiances, at the initial irradiance the Modified P&O tracked MPPT faster because the modified P&O had less settling time during changes in irradiance for different initial duty ratios. During partial shading conditions, the TCT generates more power than the S and SP topologies and modified P&O tracked the MPPT in less time. However, at the high duty ratio, the P&O and INC gave similar results, which were better than the modified P&O. It was also observed that the initial duty ratio (Di) in the MPPT algorithm had a negligible effect on TCT configuration under partial shading conditions. These findings help in various applications for PV systems.

Outlet boundary condition and mean temperature gradient effects on the minimum acoustics disturbances energy in triggering nonlinear thermoacoustic instability

In this study, we theoretically investigate the impact of outlet boundary conditions and mean temperature gradients on the maximum transient growth rate of acoustical energy and the critical energy required for triggering. Our analysis encompasses open–open and open–closed thermoacoustic systems. The theoretical models developed focus on horizontal ducts with a mean temperature jump over the heat source, employing the modified King's law. By linearizing the unsteady heat release, the nonlinear thermoacoustic equations transform into linearized-delay ones. This approach enables us to predict optimal initial perturbations for linearized-delay and nonlinear systems, corresponding to maximum transient growth rates of acoustic energy over short and long periods, respectively, thus providing insights into critical energy for triggering. We find that a closed outlet leads to higher transient energy growth and a lower critical energy for triggering compared to an open outlet. The increased mean temperature gradient has a “destructive” impact on triggering in open–open systems but a “constructive” effect in open–closed systems. Raising the mean temperature ratio generally increases the critical energy for triggering in the open–open system, whereas it decreases the critical energy in the open–closed system. The critical energy for nonlinear optimal initial perturbations is notably affected by the minimum energy of critical unstable periodic solutions, while the critical energy for linearized-delay optimal initial perturbations is closely tied to the energy level of stable periodic solutions. Due to the transient energy growth rate, the critical energy for nonlinear optimal initial perturbations is significantly lower than that for linearized-delay optimal initial perturbations.

A weakly nonlinear amplitude equation approach to the bypass transition in the two-dimensional Lamb–Oseen vortex

We analytically derive an amplitude equation for the weakly nonlinear evolution of the linearly most amplified response of a non-normal dynamical system. The development generalizes the method proposed in Ducimetière et al. (J. Fluid Mech., vol. 947, 2022, A43), in that the base flow now arbitrarily depends on time, and the operator exponential formalism for the evolution of the perturbation is not used. Applied to the two-dimensional Lamb–Oseen vortex, the amplitude equation successfully predicts the nonlinearities to weaken or reinforce the transient gain in the weakly nonlinear regime. In particular, the minimum amplitude of the linear optimal initial perturbation required for the amplitude equation to lose a solution, interpreted as the flow experiencing a bypass (subcritical) transition, is found to decay as a power law with the Reynolds number. Although with a different exponent, this is recovered in direct numerical simulations, showing a transition towards a tripolar state. The simplicity of the amplitude equation and the link made with the sensitivity formula permits a physical interpretation of nonlinear effects, in light of existing work on Landau damping and on shear instabilities. The amplitude equation also quantifies the respective contributions of the second harmonic and the spatial mean flow distortion in the nonlinear modification of the gain.

Error Evolutions and Analyses on Joint Effects of SST and SL via Intermediate Coupled Models and Conditional Nonlinear Optimal Perturbation Method

A seasonal predictability barrier has long been noticed in ENSO forecasting with numerical models. Previous studies explored the impact of seasonal optimal initial perturbation evolutions in sea surface temperature anomalies (SSTA) on ENSO forecasting using the intermediate coupled model (ICM) via the conditional nonlinear optimal perturbation (CNOP) method. In this paper, we investigate the joint effects of SSTA and sea level anomalies (SLA) from the perspective of the optimal growth initial error (OGE). After determining the four seasonal OGEs about SSTA and SLA (i.e., SSTA-OGE, SLA-OGE and Joint-OGE), we first demonstrate the patterns, evolutions and the resulting spring predictability barrier (SPB) of the above OGEs. Then, we analyze the mechanism of OGE evolutions and SPB. Finally, we conduct observing system simulation experiments to determine the best (economic) observation network. Our experimental results indicate that the ENSO evolution error induced by SSTA-OGE and Joint-OGE presents season dependency, but SLA-OGE has no impact on ENSO evolution. Moreover, Joint-OGEs induce error evolutions and the SPB with more significant intensity than SSTA-OGEs and SLA-OGEs. From mechanism analyses, the evolutions of SSTA-OGEs are mainly dominated by Bjerknes feedback. Further, the evolution dynamics of Joint-OGEs primarily contain the continuous heating between the upper ocean combined with Bjerknes feedback and thermal diffusion in response to the discharge process. In addition, comprehensive and economical sensitive areas are identified through Joint-OGE, including the central-eastern equatorial Pacific and the western and north-eastern tropical Pacific boundary, which contribute to the ENSO prediction benefits reaching 58.31% on average.

Non-modal growth of finite-amplitude disturbances in oscillatory boundary layer

The essence of sub-critical transition of oscillatory boundary-layer flows is the non-modal growth of finite-amplitude disturbances. The current understanding of the mechanisms of the orderly and bypass transitions of oscillatory boundary-layer flows is limited. The present study adopts optimisation approaches to predict the maximum energy amplification of two- and three-dimensional perturbations in response to the optimal initial disturbance with or without external forcing. A series of direct numerical simulations are also performed to compare with the results obtained from the stability analyses. In particular, the optimal initial perturbation similar to a Tollmien–Schlichting (T–S) wave yields the largest transient growth under the combined effects of the Orr mechanism and inflectional point instability. With a considerable level of two-dimensional disturbance, the vortex tube nonlinearly develops from the T–S-like wave, and then either deforms into a $\varLambda$ -vortex in the near-wall region or rolls up to the free shear region. The further burst of turbulence can follow the first pathway as K-type transition or the second one as vortex tube breakdown due to the elliptical instability. Additionally, non-modal growth can initiate the inception of streaky structures by favourable three-dimensional initial perturbations and/or forcing. The secondary instabilities responsible for the streak breakdown are classified as the varicose (symmetric) and sinuous (anti-symmetric) modes. Under a sufficiently high level of three-dimensional disturbance, the bypass transition is predominantly characterised by the formation of the sinuous mode and turbulent spots, which leads to the suppression of inflection point instability.

Using Conditional Nonlinear Optimal Perturbation to Generate Initial Perturbations in ENSO Ensemble Forecasts

AbstractUsing the latest operational version of the ENSO forecast system from the National Marine Environmental Forecasting Center (NMEFC) of China, ensemble forecasting experiments are performed for El Niño-Southern Oscillation (ENSO) events that occurred from 1997 to 2017 by generating initial perturbations of the conditional nonlinear optimal perturbation (CNOP) and Climatically relevant Singular Vector (CSV) structures. It is shown that when the initial perturbation of the leading CSV structure in the ensemble forecast of the CSVs-scheme is replaced by those of the CNOP structure, the resulted ensemble ENSO forecasts of the CNOP+CSVs-scheme tend to possess a larger spread than the forecasts obtained with the CSVs-scheme alone, leading to a better match between the root mean square error and the ensemble spread, a more reasonable Talagrand diagram and an improved Brier skill score (BSS). All these results indicate that the ensemble forecasts generated by the CNOP+CSVs-scheme can improve both the accuracy of ENSO forecasting and the reliability of the ensemble forecasting system. Therefore, ENSO ensemble forecasting should consider the effect of nonlinearity on the ensemble initial perturbations to achieve a much higher skill. It is expected that fully nonlinear ensemble initial perturbations can be sufficiently yielded to produce ensemble forecasts for ENSO, finally improving the ENSO forecast skill to the greatest possible extent. The CNOP will be a useful method to yield fully nonlinear optimal initial perturbations for ensemble forecasting.

A Linear Inverse Model of Tropical and South Pacific Climate Variability: Optimal Structure and Stochastic Forcing

AbstractA stochastically forced linear inverse model (LIM) of the combined modes of variability from the tropical and South Pacific Oceans is used to investigate the linear growth of optimal initial perturbations and to identify the spatiotemporal features of the stochastic forcing associated with the atmospheric Pacific–South American patterns 1 and 2 (PSA1 and PSA2). Optimal initial perturbations are shown to project onto El Niño–Southern Oscillation (ENSO) and South Pacific decadal oscillation (SPDO), where the inclusion of subsurface South Pacific Ocean temperature variability significantly increases the multiyear linear predictability of the deterministic system. We show that the optimal extratropical sea surface temperature (SST) precursor is associated with the South Pacific meridional mode, which takes from 7 to 9 months to linearly evolve into the final ENSO and SPDO peaks in both the observations and as simulated in an atmosphere-forced ocean model. The optimal subsurface precursor resembles its peak phase, but with a weak amplitude, representing oceanic Rossby waves in the extratropical South Pacific. The stochastic forcing is estimated as the residual by removing the deterministic dynamics from the actual tendency under a centered difference approximation. The resulting stochastic forcing time series satisfies the Gaussian white noise assumption of the LIM. We show that the PSA-like variability is strongly associated with stochastic SST forcing in the tropical and South Pacific Oceans and contributes not only to excite the optimal initial perturbations associated with ENSO and the SPDO but in general to activate the entire stochastic SST forcing, especially in austral summer.

Optimal perturbations and transition energy thresholds in boundary layer shear flows

Subcritical transition to turbulence in spatially developing boundary layer flows can be triggered efficiently by finite amplitude perturbations. In this work, we employ adjoint-based optimization to identify optimal initial perturbations in the Blasius boundary layer, culminating in the computation of the subcritical transition critical energy threshold and the associated fully localized critical optimum in a spatially extended configuration, the so called minimal seed. By dynamically rescaling the variables with the local boundary layer thickness, we show that the identified edge trajectory approaches the same attracting phase space region as previously reported edge trajectories, and reaches the region more efficiently.

Optimal perturbations for controlling the growth of a Rayleigh–Taylor instability

A discrete adjoint-based method is employed to control multi-mode Rayleigh–Taylor (RT) instabilities via strategic manipulation of the initial interfacial perturbations. We seek to find to what extent mixing and growth can be enhanced at late stages of the instability and which modes are targeted to achieve this. Three objective functions are defined to quantify RT mixing and growth: (i) variance of mole fraction, (ii) a kinetic energy norm based on the vertical velocity and (iii) variations of mole fraction with respect to the unperturbed initial state. The sensitivity of these objective functions to individual amplitudes of the initial perturbations are studied at various stages of the RT instability. The most sensitive wavenumber during the early stages of the instability closely matches the most unstable wavenumber predicted by linear stability theory. It is also shown that randomly changing the initial perturbations has little effect at early stages, but results in large variations in both RT growth and its sensitivity at later times. The sensitivity obtained from the adjoint solution is employed in gradient-based optimization to both suppress and enhance the objective functions. The adjoint-based optimization framework was capable of completely suppressing mixing by shifting all of the interfacial perturbation energy to the highest modes such that diffusion dominates. The optimal initial perturbations for enhancing the objective functions were found to have a broadband spectrum resulting in non-trivial coupling between modes and depends on the particular objective function being optimized. The objective functions were increased by as much as a factor of nine in the self-similar late-stage growth regime compared to an interface with a uniform distribution of modes, corresponding to a 32% increase in the bubble growth parameter and 54% increase in the mixing width. It was also found that the interfacial perturbations optimized at early stages of the instability are unable to predict enhanced mixing at later times, and thus optimizing late-time multi-mode RT instabilities requires late-time sensitivity information. Finally, it was found that the optimized distribution of interfacial perturbations obtained from two-dimensional simulations was capable of enhancing the objective functions in three-dimensional flows. As much as 51% and 99% enhancement in the bubble growth parameter and mixing width, respectively, was achieved, even greater than what was reached in two dimensions.

Fingering instability on curved substrates: optimal initial film and substrate perturbations

We investigate the stability of a thin Newtonian fluid spreading on a horizontal cylinder under the action of gravity. The capillary ridge forming at the advancing front is known to be unstable with respect to spanwise perturbations, resulting in the formation of fingers. In contrast to the classic case of a flow over an inclined plane, the gravity components along a cylindrical substrate vary in space and the draining flow is time-dependent, making a modal stability analysis inappropriate. A linear optimal transient growth analysis is instead performed to find the optimal spanwise wavenumber. We not only consider the optimal perturbations of the initial film thickness, as commonly done in the literature, but also the optimal topographical perturbations of the substrate, which are of significant practical relevance. We found that, in both cases, the optimal gains are obtained when the perturbation structures are the least affected by the time horizon. The optimal spanwise wavenumber is found to be dependent on the front location, due to the dependence of the characteristic length of the capillary ridge on its polar location.

A useful approach to sensitivity and predictability studies in geophysical fluid dynamics: conditional non-linear optimal perturbation.

In atmospheric and oceanic studies, it is important to investigate the uncertainty of model solutions. The conditional non-linear optimal perturbation (CNOP) method is useful for addressing the uncertainty. This paper reviews the development of the CNOP method and its computational aspects in recent years. Specifically, the CNOP method was first proposed to investigate the effects of the optimal initial perturbation on atmosphere and ocean model results. Then, it was extended to explore the influences of the optimal parameter perturbation, model tendency perturbation and boundary condition perturbation. To obtain solutions to these optimal perturbations, four kinds of optimization approaches were developed: the adjoint-based method, the adjoint-free method, the intelligent optimization method and the unconstrained optimization method. We illustrate the calculation process of each method and its advantages and disadvantages. Then, taking the Zebiak–Cane model as an example, we compare the CNOPs related to initial conditions (CNOP-Is) calculated by the above four methods. It was found that the dominant structures of the CNOP-Is for different methods are similar, although some differences in details exist. Finally, we discuss the necessity and possible direction for designing a more effective optimization approach related to the CNOP in the future.

Optimal mixing in three-dimensional plane Poiseuille flow at high Péclet number

We consider a passive zero-mean scalar field organised into two layers of different concentrations in a three-dimensional plane channel flow subjected to a constant along-stream pressure gradient. We employ a nonlinear direct-adjoint-looping method to identify the optimal initial perturbation of the velocity field with given initial energy which yields ‘maximal’ mixing by a target time horizon, where maximal mixing is defined here as the minimisation of the spatially integrated variance of the concentration field. We verify in three-dimensional flows the conjecture by Foures et al. (J. Fluid Mech., vol. 748, 2014, pp. 241–277) that the initial perturbation which maximises the time-averaged energy gain of the flow leads to relatively weak mixing, and is qualitatively different from the optimal initial ‘mixing’ perturbation which exploits classical Taylor dispersion. We carry out the analysis for two different Reynolds numbers ($Re=U_{m}h/\unicode[STIX]{x1D708}=500$ and $Re=3000$, where $U_{m}$ is the maximum flow speed of the unperturbed flow, $h$ is the channel half-depth and $\unicode[STIX]{x1D708}$ is the kinematic viscosity of the fluid) demonstrating that this key finding is robust with respect to the transition to turbulence. We also identify the initial perturbations that minimise, at chosen target times, the ‘mix-norm’ of the concentration field, i.e. a Sobolev norm of negative index in the class introduced by Mathew et al. (Physica D, vol. 211, 2005, pp. 23–46). We show that the ‘true’ variance-based mixing strategy can be successfully and practicably approximated by the mix-norm minimisation since we find that the mix-norm-optimal initial perturbations are far less sensitive to changes in the target time horizon than their optimal variance-minimising counterparts.

Transient perturbation growth of vortex rings

Perturbations to a vortex ring with largest transient energy growth are calculated and applied to drive the vortex ring to evolve. Optimal initial perturbations appear at azimuthal wave numbers β = 0 and β = 9. The former was described as ‘axial flow’ and is not related to the breakdown of the ring, while the latter is critical to the ring breakdown process as observed in previous direct numerical simulation (DNS) with random initial noise input. In the present work, DNS is conducted to study the nonlinear development of the optimal perturbations, and good agreement with past studies is achieved.

Optimal Initial Excitations of Decadal Modification of the Atlantic Meridional Overturning Circulation under the Prescribed Heat and Freshwater Flux Boundary Conditions

AbstractWithin a three-dimensional ocean circulation model, the nonlinear optimal initial perturbations (NOIP) of sea surface salinity (SSS) and sea surface temperature (SST) to excite variability in the Atlantic meridional overturning circulation (AMOC) were obtained under prescribed heat and freshwater flux boundary conditions, using the conditional nonlinear optimal perturbation (CNOP) method. After 10 years, the optimal SSS and SST perturbations lead to reductions of the AMOC by 3.6 and 2.5 Sv (1 Sv = 106 m3 s−1), respectively, followed by multidecadal oscillations with a period of about 50 years. During the first 30 years, nonlinear processes have an important influence on the AMOC strength: convection strengthens the AMOC during years 0–2, zonal density advection promotes the slowdown of the AMOC during years 7–20, and meridional density advection inhibits the slowdown of meridional velocities in the upper ocean during years 5–18. The linear optimal initial perturbation (LOIP) was also computed using the first singular vector (FSV) method. For SSS perturbations with an amplitude of 0.5 psu, the LOIP will cause an underestimation of the amplitude of the multidecadal AMOC variability by about 1 Sv, compared to that induced by the NOIP. This underestimation will become more significant as the amplitudes of SSS perturbations increase.

Optimal suppression of flow perturbations using boundary control

Boundary perturbations are considered as flow control forcing and their distributions are optimised to suppress transient energy growth induced by the most energetic disturbances in the domain. For a given control cost (square integration of the control forcing), the optimal control calculated from the proposed optimisation algorithm is proved to be unique. For small values of control cost, a sensitivity solution is obtained and its distribution indicates the sensitivity of perturbation energy on boundary control. For larger control cost, the distribution of the optimal control approaches the stablest mode of a direct-adjoint operator and tends to be grid-to-grid oscillatory. A controllability analysis is further conducted to identify the uncontrollable component of perturbations in the domain. This work underpins the recently thriving linear feed-back flow control investigations, most of which use empirically distributed control actuators, in terms of choosing the location and magnitude of the control forcing and evaluating the maximum control effect. Two case studies are conducted to demonstrate the proposed algorithm; in a stenotic flow, the optimised wall boundary control is observed to suppress over 95% of the transient energy growth induced by the global optimal initial perturbation; in the Batchelor vortex flow, the optimal inflow control can effectively suppress the spiral vortex breakdown induced by the development of initial perturbations.

Three-Dimensional Structure of Optimal Nonlinear Excitation for Decadal Variability of the Thermohaline Circulation

AbstractThe decadal variability of the North Atlantic thermohaline circulation (THC) is investigated within a three-dimensional ocean circulation model using the conditional nonlinear optimal perturbation method. The results show that the optimal initial perturbations of temperature and salinity exciting the strongest decadal THC variations have similar structures: the perturbations are mainly in the northwestern basin at a depth ranging from −1500 to −3000 m. These temperature and salinity perturbations act as the optimal precursors for future modifications of the THC, highlighting the importance of observations in the northwestern basin to monitor the variations of temperature and salinity at depth. The decadal THC variation in the nonlinear model initialized by the optimal salinity perturbations is much stronger than that caused by the optimal temperature perturbations, indicating that salinity variations might play a relatively important role in exciting the decadal THC variability. Moreover, the deca...

Modal and non-modal stabilities of flow around a stack of plates

Modal and non-modal stabilities of flow around a stack of flat plates are investigated by means of asymptotic stability and transient growth analyses respectively. It is observed that over the parameters considered, both the base flow and the stabilities vary as a function of ReW2/(W−1)2, i.e. the product of the Reynolds number and the square of the expansion ratio of the stack. The most unstable modes are found to be located downstream of the recirculation bubble while the global optimal initial perturbations (resulting in maximum energy growth over the entire domain) and the weighted optimal initial perturbations (resulting in maximum energy growth in the close downstream region of the stack) concentrate around the stack end owing to the Orr mechanism. In direct numerical simulations (DNS) of the base flow initially perturbed by the modes, it is noticed that the weighted optimal initial perturbation induces periodic vortex shedding downstream of the stack much faster than the most unstable mode. This observation suggests that the widely reported vortex shedding in flow around a stack of plates, e.g. in thermoacoustic devices, is associated with perturbations around the stack end.

Effects of base flow modifications on noise amplifications: flow past a backward-facing step

Amplifications of flow past a backward-facing step with respect to optimal inflow and initial perturbations are investigated at Reynolds number 500. Two mechanisms of receptivity to inflow noise are identified: the bubble-induced inflectional point instability and the misalignment effect downstream of the secondary bubble. Further development of the misalignment results in decay of perturbations from $x=28$ onwards (the step is located at $x=0$), as has been observed in previous non-normality studies (Blackburn et al., J. Fluid Mech., vol. 603, 2008, pp. 271–304), and eventually limits the receptivity. The receptivity is found to be maximized at an inflow perturbation frequency of ${\it\omega}=0.50$ and a spanwise wavenumber of ${\it\beta}=0$, where the inflow noise takes full advantage of both mechanisms and is amplified over two orders of magnitude in terms of the velocity magnitude. In direct numerical simulations (DNS) of the flow perturbed by optimal or random inflow noise, vortex shedding, flapping of bubbles, three-dimensionality and turbulence are observed in succession as the magnitude of the inflow noise increases. Similar features of linear and nonlinear receptivity are observed at higher Reynolds numbers. The Strouhal number of the bubble flapping is 0.08, at which the receptivity to inflow noise reaches a maximum. This Strouhal number is close to reported values extracted from DNS or large eddy simulations (LES) at larger Reynolds numbers (Le et al., J. Fluid Mech., vol. 330, 1997, pp. 349–374; Kaiktsis et al., J. Fluid Mech., vol. 321, 1996, pp. 157–187; Métais, New Trends in Turbulence, 2001, Springer; Wee et al., Phys. Fluids, vol. 16, 2004, pp. 3361–3373). Methods to further clarify the mechanisms of receptivity and to suppress the noise amplifications by modifying the base flow using a linearly optimal body force are proposed. It is observed that the mechanisms of optimal noise amplification are fully revealed by the distribution of the base flow modification, which weakens the bubble instabilities and misalignment effects and subsequently reduces the receptivity significantly. Comparing the base flow modifications with respect to amplifications of inflow and initial perturbations, it is found that the maximum receptivity to initial perturbations is highly correlated with the receptivity to inflow noise at the optimal frequency ${\it\omega}=0.50$, and the correlation reduces as the inflow frequency deviates from this optimal value.

Optimal mixing in two-dimensional plane Poiseuille flow at finite Péclet number

AbstractWe consider the nonlinear optimisation of the mixing of a passive scalar, initially arranged in two layers, in a two-dimensional plane Poiseuille flow at finite Reynolds and Péclet numbers, below the linear instability threshold. We use a nonlinear-adjoint-looping approach to identify optimal perturbations leading to maximum time-averaged energy as well as maximum mixing in a freely evolving flow, measured through the minimisation of either the passive scalar variance or the so-called mix-norm, as defined by Mathew, Mezić & Petzold (Physica D, vol. 211, 2005, pp. 23–46). We show that energy optimisation appears to lead to very weak mixing of the scalar field whereas the optimal mixing initial perturbations, despite being less energetic, are able to homogenise the scalar field very effectively. For sufficiently long time horizons, minimising the mix-norm identifies optimal initial perturbations which are very similar to those which minimise scalar variance, demonstrating that minimisation of the mix-norm is an excellent proxy for effective mixing in this finite-Péclet-number bounded flow. By analysing the time evolution from initial perturbations of several optimal mixing solutions, we demonstrate that our optimisation method can identify the dominant underlying mixing mechanism, which appears to be classical Taylor dispersion, i.e. shear-augmented diffusion. The optimal mixing proceeds in three stages. First, the optimal mixing perturbation, energised through transient amplitude growth, transports the scalar field across the channel width. In a second stage, the mean flow shear acts to disperse the scalar distribution leading to enhanced diffusion. In a final third stage, linear relaxation diffusion is observed. We also demonstrate the usefulness of the developed variational framework in a more realistic control case: mixing optimisation by prescribed streamwise velocity boundary conditions.

Transient growth of Ekman-Couette flow

Coriolis force effects on shear flows are important in geophysical and astrophysical contexts. We report a study on the linear stability and the transient energy growth of the plane Couette flow with system rotation perpendicular to the shear direction. External rotation causes linear instability. At small rotation rates, the onset of linear instability scales inversely with the rotation rate and the optimal transient growth in the linearly stable region is slightly enhanced ∼Re2. The corresponding optimal initial perturbations are characterized by roll structures inclined in the streamwise direction and are twisted under external rotation. At large rotation rates, the transient growth is significantly inhibited and hence linear stability analysis is a reliable indicator for instability.